Weighted and shifted BDF3 methods on variable grids for a parabolic problem
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As is well known, the stability of the 3-step backward differentiation formula (BDF3) on variable grids for a parabolic problem is analyzed in [Calvo and Grigorieff, BIT. 42 (2002) 689–701] under the condition r_k:=τ_k/τ_k-1<1.199, where r_k is the adjacent time-step ratio. In this work, we establish the spectral norm inequality, which can be used to give a upper bound for the inverse matrix. Then the BDF3 scheme is unconditionally stable under a new condition r_k≤ 1.405. Meanwhile, we show that the upper bound of the ratio r_k is less than √(3) for BDF3 scheme. In addition, based on the idea of [Wang and Ruuth, J. Comput. Math. 26 (2008) 838–855; Chen, Yu, and Zhang, arXiv:2108.02910], we design a weighted and shifted BDF3 (WSBDF3) scheme for solving the parabolic problem. We prove that the WSBDF3 scheme is unconditionally stable under the condition r_k≤ 1.771, which is a significant improvement for the maximum time-step ratio. The error estimates are obtained by the stability inequality. Finally, numerical experiments are given to illustrate the theoretical results.
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